25.63 HallSubgroup

HallSubgroup( G, n )
HallSubgroup( G, L )

Let G be an ag group. Then HallSubgroup returns a pi-Hall-subgroup of G for the set pi of all prime divisors of the integer n or the join pi of all prime divisors of the integers of L.

The Hall-subgroup is constructed using Glasby's algorithm (see Gla87), which descends along an elementary abelian series for G and constructs complements in the coprime case (see CoprimeComplement). If no such series is known for G the function uses ElementaryAbelianSeries (see ElementaryAbelianSeries) in order to construct such a series for G.

    gap> HallSubgroup( s4, 2 );
    Subgroup( s4, [ a, c, d ] )
    gap> HallSubgroup( s4, [ 3 ] );
    Subgroup( s4, [ b ] )
    gap> z5 := CyclicGroup( AgWords, 5 );
    Group( c5 )
    gap> DirectProduct( s4, z5 );
    Group( a1, a2, a3, a4, b )
    gap> HallSubgroup( last, [ 5, 3 ] );
    Subgroup( Group( a1, a2, a3, a4, b ), [ a2, b ] ) 

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